Dust production from collisions in extrasolar planetary systems. The inner beta Pictoris disc

Dust particles observed in extrasolar planetary discs originate from undetectable km-sized bodies but this valuable information remains uninteresting if the theoretical link between grains and planetesimals is not properly known. We outline in this paper a numerical approach we developed in order to address this issue for the case of dust producing collisional cascades. The model is based on a particle-in-a-box method. We follow the size distribution of particles over eight orders of magnitude in radius taking into account fragmentation and cratering according to different prescriptions. Particular attention is paid to the smallest particles, close to the radiation pressure induced cut-off size R_pr, which are placed on highly eccentric orbits by the stellar radiation pressure. We applied our model to the case of the inner (<10 AU) β Pictoris disc, in order to quantitatively derive the population of progenitors needed to produce the small amount of dust observed in this region (∼= 10²² g). Our simulations show that the collisional cascade from kilometre-sized bodies to grains significantly departs from the classical dN∝ R^-3.5 dR power law: the smallest particles (R∼= R_pr) are strongly depleted while an overabundance of grains with size ∼ 2R_pr and a drop of grains with size ∼ 100 R_pr develop regardless of the disc's dynamical excitation, R_pr and initial surface density. However, the global dust to planetesimal mass ratio remains close to its dN∝ R^-3.5 dR value. Our rigorous approach thus confirms the depletion in mass in the inner β Pictoris disc initially inferred from questionable assumptions. We show moreover that collisions are a sufficient source of dust in the inner β Pictoris disc. They are actually unavoidable even when considering the alternative scenario of dust production by slow evaporation of km-sized bodies. We obtain an upper limit of ∼ 0.1 M_⊕ for the total disc mass below 10 AU. This upper limit is not consistent with the independent mass estimate (at least 15 M_⊕) in the frame of the Falling Evaporating Bodies (FEB) scenario explaining the observed transient features activity. Furthermore, we show that the mass required to sustain the FEB activity implies a so important mass loss that the phenomena should naturally end in less than 1 Myr, namely in less than one twentieth the age of the star (at least 2 × 10⁷ years). In conclusion, these results might help converge towards a coherent picture of the inner β Pictoris system: a low-mass disc of collisional debris leftover after the possible formation of planetary embryos, a result which would be coherent with the estimated age of the system.